Journal articles: 'Euler Equation - Numerical Solution'

1

Čiegis, Raimondas. "NUMERICAL SOLUTION OF HYPERBOLIC HEAT CONDUCTION EQUATION." Mathematical Modelling and Analysis 14, no. 1 (March 31, 2009): 11–24. http://dx.doi.org/10.3846/1392-6292.2009.14.11-24.

Full text

Abstract:

Hyperbolic heat conduction problem is solved numerically. The explicit and implicit Euler schemes are constructed and investigated. It is shown that the implicit Euler scheme can be used to solve efficiently parabolic and hyperbolic heat conduction problems. This scheme is unconditionally stable for both problems. For many integration methods strong numerical oscillations are present, when the initial and boundary conditions are discontinuous for the hyperbolic problem. In order to regularize the implicit Euler scheme, a simple linear relation between time and space steps is proposed, which automatically introduces sufficient amount of numerical viscosity. Results of numerical experiments are presented.

APA, Harvard, Vancouver, ISO, and other styles

2

Siddiqui, Maryam, Mhamed Eddahbi, and Omar Kebiri. "Numerical Solutions of Stochastic Differential Equations with Jumps and Measurable Drifts." Mathematics 11, no. 17 (August 31, 2023): 3755. http://dx.doi.org/10.3390/math11173755.

Full text

Abstract:

This paper deals with numerical analysis of solutions to stochastic differential equations with jumps (SDEJs) with measurable drifts that may have quadratic growth. The main tool used is the Zvonkin space transformation to eliminate the singular part of the drift. More precisely, the idea is to transform the original SDEJs to standard SDEJs without singularity by using a deterministic real-valued function that satisfies a second-order differential equation. The Euler–Maruyama scheme is used to approximate the solution to the equations. It is shown that the rate of convergence is 12. Numerically, two different methods are used to approximate solutions for this class of SDEJs. The first method is the direct approximation of the original equation using the Euler–Maruyama scheme with specific tests for the evaluation of the singular part at simulated values of the solution. The second method consists of taking the inverse of the Euler–Maruyama approximation for Zvonkin’s transformed SDEJ, which is free of singular terms. Comparative analysis of the two numerical methods is carried out. Theoretical results are illustrated and proved by means of an example.

APA, Harvard, Vancouver, ISO, and other styles

3

Wang, Qi, Huabin Chen, and Chenggui Yuan. "A Note on Exponential Stability for Numerical Solution of Neutral Stochastic Functional Differential Equations." Mathematics 10, no. 6 (March 9, 2022): 866. http://dx.doi.org/10.3390/math10060866.

Full text

Abstract:

This paper examines the numerical solutions of the neutral stochastic functional differential equation. This study establishes the discrete stochastic Razumikhin-type theorem to investigate the exponential stability in the mean square sense of the Euler–Maruyama numerical solution to this equation. In addition, the Borel–Cantelli lemma and the stochastic analysis theory are incorporated to discuss the almost sure exponential stability for this numerical solution of such equations.

APA, Harvard, Vancouver, ISO, and other styles

4

Wang, Jinhuan, Yicheng Pang, and Yu Zhang. "Limits of Solutions to the Isentropic Euler Equations for van der Waals Gas." International Journal of Nonlinear Sciences and Numerical Simulation 20, no. 3-4 (May 26, 2019): 461–73. http://dx.doi.org/10.1515/ijnsns-2018-0263.

Full text

Abstract:

AbstractIn this paper, we consider limit behaviors of Riemann solutions to the isentropic Euler equations for a non-ideal gas (i.e. van der Waals gas) as the pressure vanishes. Firstly, the Riemann problem of the isentropic Euler equations for van der Waals gas is solved. Then it is proved that, as the pressure vanishes, any Riemann solution containing two shock waves to the isentropic Euler equation for van der Waals gas converges to the delta shock solution to the transport equations and any Riemann solution containing two rarefaction waves tends to the vacuum state solution to the transport equations. Finally, some numerical simulations completely coinciding with the theoretical analysis are demonstrated.

APA, Harvard, Vancouver, ISO, and other styles

5

Blaszczyk, Tomasz, and Mariusz Ciesielski. "Numerical Solution of Euler-Lagrange Equation with Caputo Derivatives." Advances in Applied Mathematics and Mechanics 9, no. 1 (October 11, 2016): 173–85. http://dx.doi.org/10.4208/aamm.2015.m970.

Full text

Abstract:

AbstractIn this paper the fractional Euler-Lagrange equation is considered. The fractional equation with the left and right Caputo derivatives of orderα∈ (0,1] is transformed into its corresponding integral form. Next, we present a numerical solution of the integral form of the considered equation. On the basis of numerical results, the convergence of the proposed method is determined. Examples of numerical solutions of this equation are shown in the final part of this paper.

APA, Harvard, Vancouver, ISO, and other styles

6

Shieh, C. F., and R. A. Delaney. "An Accurate and Efficient Euler Solver for Three-Dimensional Turbomachinery Flows." Journal of Turbomachinery 109, no. 3 (July 1, 1987): 346–53. http://dx.doi.org/10.1115/1.3262112.

Full text

Abstract:

Accurate and efficient Euler equation numerical solution techniques are presented for analysis of three-dimensional turbomachinery flows. These techniques include an efficient explicit hopscotch numerical scheme for solution of the three-dimensional time-dependent Euler equations and an O-type body-conforming grid system. The hopscotch scheme is applied to the conservative form of the Euler equations written in general curvilinear coordinates. The grid is constructed by stacking from hub to shroud two-dimensional O-type grids on equally spaced surfaces of revolution. Numerical solution results for two turbine cascades are presented and compared with experimental data to demonstrate the accuracy of the analysis method.

APA, Harvard, Vancouver, ISO, and other styles

7

Sarhan, Falah, and LIU JICHENG. "Euler-Maruyama approximation of backward doubly stochastic differential delay equations." International Journal of Applied Mathematical Research 5, no. 3 (July 25, 2016): 146. http://dx.doi.org/10.14419/ijamr.v5i3.6358.

Full text

Abstract:

In this paper, we attempt to introduce a new numerical approach to solve backward doubly stochastic differential delay equation ( shortly-BDSDDEs ). In the beginning, we present some assumptions to get the numerical scheme for BDSDDEs, from which we prove important theorem. We use the relationship between backward doubly stochastic differential delay equations and stochastic controls by interpreting BDSDDEs as some stochastic optimal control problems, to solve the approximated BDSDDEs and we prove that the numerical solutions of backward doubly stochastic differential delay equation converge to the true solution under the Lipschitz condition.

APA, Harvard, Vancouver, ISO, and other styles

8

CHINVIRIYASIT, W., and S. CHINVIRIYASIT. "NUMERICAL SOLUTION OF NONLINEAR EQUATION TO COMBINED DETERMINISTIC AND NARROW-BAND RANDOM EXCITATION." International Journal of Modern Physics B 22, no. 21 (August 20, 2008): 3655–75. http://dx.doi.org/10.1142/s0217979208048528.

Full text

Abstract:

The Duffing oscillator to combined deterministic and narrow-band random excitation, which is a nonlinear equation, is studied and solved numerically using three numerical methods based on finite difference schemes. Method 1, the well-known Euler method, is an explicit method; Method 2 is an implicit first-order method which does not bring contrived chaos into the solution; and Method 3 is based on two first-order methods which is second-order method and is chaos-free. In a series of numerical experiments, it is seen that the proposed methods have superior stability properties to those of the well-known Euler and fourth-order Runge-Kutta methods to which they are compared. When extended to the numerical solution of Duffing oscillator to combined deterministic and narrow-band random excitation, the developed methods give the correct steady-state solutions compared with the literature.

APA, Harvard, Vancouver, ISO, and other styles

9

Zhang, X., X. X. Chen, and C. L. Morfey. "Acoustic Radiation from a Semi-Infinite Duct With a Subsonic Jet." International Journal of Aeroacoustics 4, no. 1-2 (January 2005): 169–84. http://dx.doi.org/10.1260/1475472053730075.

Full text

Abstract:

The radiation of high-order spinning modes from a semi-infinite exhaust duct is studied numerically. The issues involved have applications to noise radiation from the exhaust duct of an aircraft engine. The numerical method is based on solutions of linearised Euler equations (LEE) for propagation in the duct and near field, and the acoustic analogy for far field radiation. A 2.5D formulation of a linearised Euler equation model is employed to accommodate a single spinning mode propagating over an axisymmetric mean flow field. In the solution process, acoustic waves are admitted into the propagation area surrounding the exit of an axisymmetric duct and its immediate downstream area. The wave admission is realised through an absorbing non-reflecting boundary treatment, which admits incoming waves and damps spurious waves generated by the numerical solutions. The wave propagation is calculated through solutions of linearised Euler equations, using an optimised prefactored compact scheme for spatial discretisation. Far field directivity is estimated by solving the Ffowcs Williams-Hawkings equations. The far field prediction is compared with analytic solutions with good agreement.

APA, Harvard, Vancouver, ISO, and other styles

10

Khaldi, Rabah, and Assia Guezane-Lakoud. "On generalized nonlinear Euler-Bernoulli Beam type equations." Acta Universitatis Sapientiae, Mathematica 10, no. 1 (August 1, 2018): 90–100. http://dx.doi.org/10.2478/ausm-2018-0008.

Full text

Abstract:

Abstract This paper is devoted to the study of a nonlinear Euler-Bernoulli Beam type equation involving both left and right Caputo fractional derivatives. Differently from the approaches of the other papers where they established the existence of solution for the linear Euler-Bernoulli Beam type equation numerically, we use the lower and upper solutions method with some new results on the monotonicity of the right Caputo derivative. Furthermore, we give the explicit expression of the upper and lower solutions. A numerical example is given to illustrate the obtained results.

APA, Harvard, Vancouver, ISO, and other styles

11

Singh, Sudhansu, and Mohapatra Dinakrushna. "Solution of the reactor point kinetics equations by MATLAB computing." Nuclear Technology and Radiation Protection 30, no. 1 (2015): 11–17. http://dx.doi.org/10.2298/ntrp1501011s.

Full text

Abstract:

The numerical solution of the point kinetics equations in the presence of Newtonian temperature feedback has been a challenging issue for analyzing the reactor transients. Reactor point kinetics equations are a system of stiff ordinary differential equations which need special numerical treatments. Although a plethora of numerical intricacies have been introduced to solve the point kinetics equations over the years, some of the simple and straightforward methods still work very efficiently with extraordinary accuracy. As an example, it has been shown recently that the fundamental backward Euler finite difference algorithm with its simplicity has proven to be one of the most effective legacy methods. Complementing the back-ward Euler finite difference scheme, the present work demonstrates the application of ordinary differential equation suite available in the MATLAB software package to solve the stiff reactor point kinetics equations with Newtonian temperature feedback effects very effectively by analyzing various classic benchmark cases. Fair accuracy of the results implies the efficient application of MATLAB ordinary differential equation suite for solving the reactor point kinetics equations as an alternate method for future applications.

APA, Harvard, Vancouver, ISO, and other styles

12

Lu, Chang-Na, Sheng-Xiang Chang, Luo-Yan Xie, and Zong-Guo Zhang. "Generation and solutions to the time-space fractional coupled Navier-Stokes equations." Thermal Science 24, no. 6 Part B (2020): 3899–905. http://dx.doi.org/10.2298/tsci2006899l.

Full text

Abstract:

In this paper, a Lagrangian of the coupled Navier-Stokes equations is proposed based on the semi-inverse method. The fractional derivatives in the sense of Riemann-Liouville definition are used to replace the classical derivatives in the Lagrangian. Then the fractional Euler-Lagrange equation can be derived with the help of the fractional variational principles. The Agrawal?s method is devot?ed to lead to the time-space fractional coupled Navier-Stokes equations from the above Euler-Lagrange equation. The solution of the time-space fractional coupled Navier-Stokes equations is obtained by means of RPS algorithm. The numerical results are presented by using exact solutions.

APA, Harvard, Vancouver, ISO, and other styles

13

Habibah, Ummu, Mohamad Handri Tuloli, Viva Rimanada, and Tomas Goncalves Ferreira. "Penyelesaian Numerik Masalah Syarat Batas Robin pada Persamaan Diferensial Cauchy-Euler." MAJAMATH: Jurnal Matematika dan Pendidikan Matematika 3, no. 1 (March 5, 2020): 32–40. http://dx.doi.org/10.36815/majamath.v3i1.615.

Full text

Abstract:

This research studied how the numerical solution of the Cauchy-Euler differential equation with Robin boundary conditions. There were several numerical methods that can be used to get the numerical solution of a boundary value problem, namely the finite-difference method, the shooting method, the collocation method, and others. In this study, the numerical solution of Robin's boundary condition problem was obtained by the center finite-difference and the shooting methods. From the two methods, the numerical error was compared to the exact solution. The simulation results shown that the shooting method produces a better numerical solution for approximating the completion of the Cauchy-Euler differential equation than the finite-difference method since it produced smaller numerical errors.

APA, Harvard, Vancouver, ISO, and other styles

14

Smaoui, Nejib. "Controlling the dynamics of Burgers equation with a high-order nonlinearity." International Journal of Mathematics and Mathematical Sciences 2004, no. 62 (2004): 3321–32. http://dx.doi.org/10.1155/s0161171204404116.

Full text

Abstract:

We investigate analytically as well as numerically Burgers equation with a high-order nonlinearity (i.e.,ut=νuxx−unux+mu+h(x)). We show existence of an absorbing ball inL2[0,1]and uniqueness of steady state solutions for all integern≥1. Then, we use an adaptive nonlinear boundary controller to show that it guarantees global asymptotic stability in time and convergence of the solution to the trivial solution. Numerical results using Chebychev collocation method with backward Euler time stepping scheme are presented for both the controlled and the uncontrolled equations illustrating the performance of the controller and supporting the analytical results.

APA, Harvard, Vancouver, ISO, and other styles

15

ШайдуровВ.В., ШайдуровВ В., and ЧередниченкоО М. ЧередниченкоО.М. "Semi-Lagrangian approximations of the convection operator in symmetric form." Вычислительные технологии, no. 3 (June 21, 2023): 101–16. http://dx.doi.org/10.25743/ict.2023.28.3.007.

Full text

Abstract:

Рассмотрены два полулагранжевых численных метода для одномерного (по пространству) уравнения переноса с оператором в симметричной форме: эйлероволагранжев и лагранжево-эйлеров. Оба метода свободны от ограничения Куранта на соотношение шагов по времени и пространству. Причем во втором методе достигнут второй порядок аппроксимации для гладких решений и продемонстрировано отсутствие численной вязкости для разрывных решений. Purpose. The purpose of the study is the development and comparison of two numerical semi-Lagrangian methods with fulfillment of the conservation law at a discrete level. The approach is applied for the transport equation in the symmetric form, reflecting the law of conservation for the square of the transferred substance. The article presents the Euler – Lagrangian method, built on a rectangular difference grid that uses local values of characteristics to calculate the coefficients of difference equations. Lagrangian – Euler method is built on a spatial non-uniform grid obtained by crossing the characteristic trajectories of the equation with lines in time. Methodology. The integro-interpolation method is applied to derive approximations for the differential operator which allowed obtaining simple formulas connecting values of the grid function at the neighboring layers in time. Numerical calculations of characteristic trajectories are held by the Euler method or the Runge – Kutta method of the second order, depending on the required accuracy. Findings. Numerical methods with the mentioned properties are developed and numerically confirmed, convergence and discrete conservation laws for them are mathematically proved. The first order convergence for both time and space is proved for the Euler – Lagrange method. The second order convergence also in time and space is proved for the Lagrange – Euler method. Originality/value. The Euler – Lagrange and Lagrange – Euler methods for the numerical solution of the convection equation are developed. These methods induce differential conservation law at discrete level. The first and the second order of convergence correspondingly are mathematically proved for them. The Lagrange – Euler method has showed two improved aspects: firstly, it has greater order of convergence than the Euler – Lagrange one and secondly, it allows solving problems with the discontinuous solutions without smoothing them.

APA, Harvard, Vancouver, ISO, and other styles

16

Fiori, Simone. "Model Formulation Over Lie Groups and Numerical Methods to Simulate the Motion of Gyrostats and Quadrotors." Mathematics 7, no. 10 (October 10, 2019): 935. http://dx.doi.org/10.3390/math7100935.

Full text

Abstract:

The present paper recalls a formulation of non-conservative system dynamics through the Lagrange–d’Alembert principle expressed through a generalized Euler–Poincaré form of the system equation on a Lie group. The paper illustrates applications of the generalized Euler–Poincaré equations on the rotation groups to a gyrostat satellite and a quadcopter drone. The numerical solution of the dynamical equations on the rotation groups is tackled via a generalized forward Euler method and an explicit Runge–Kutta integration method tailored to Lie groups.

APA, Harvard, Vancouver, ISO, and other styles

17

MEIRON, D. I., D. W. MOORE, and D. I. PULLIN. "On steady compressible flows with compact vorticity; the compressible Stuart vortex." Journal of Fluid Mechanics 409 (April 25, 2000): 29–49. http://dx.doi.org/10.1017/s0022112099007752.

Full text

Abstract:

Numerical and analytical solutions to the steady compressible Euler equations corresponding to a compressible analogue of the linear Stuart vortex array are presented. These correspond to a homentropic continuation, to finite Mach number, of the Stuart solution describing a linear vortex array in an incompressible fluid. The appropriate partial differential equations describing the flow correspond to the compressible homentropic Euler equations in two dimensions, with a prescribed vorticity–density–streamfunction relationship. In order to construct a well-posed problem for this continuation, it was found, unexpectedly, to be necessary to introduce an eigenvalue into the vorticity–density–streamfunction equation. In the Rayleigh–Janzen expansion of solutions in even powers of the free-stream Mach number M∞, this eigenvalue is determined by a solvability condition. Accurate numerical solution by both finite-difference and spectral methods are presented for the compressible Stuart vortex, over a range of M∞, and of a parameter corresponding to a confined mass-flow rate. These also confirm the nonlinear eigenvalue character of the governing equations. All solution branches followed numerically were found to terminate when the maximum local Mach number just exceeded unity. For one such branch we present evidence for the existence of a very small range of M∞ over which smooth transonic shock-free flow can occur.

APA, Harvard, Vancouver, ISO, and other styles

18

BERNACKI, MARC, and SERGE PIPERNO. "A DISSIPATION-FREE TIME-DOMAIN DISCONTINUOUS GALERKIN METHOD APPLIED TO THREE-DIMENSIONAL LINEARIZED EULER EQUATIONS AROUND A STEADY-STATE NON-UNIFORM INVISCID FLOW." Journal of Computational Acoustics 14, no. 04 (December 2006): 445–67. http://dx.doi.org/10.1142/s0218396x0600313x.

Full text

Abstract:

We present in this paper a time-domain discontinuous Galerkin dissipation-free method for the transient solution of the three-dimensional linearized Euler equations around a steady-state solution. In the general context of a nonuniform supporting flow, we prove, using the well-known symmetrization of Euler equations, that some aeroacoustic energy satisfies a balance equation with source term at the continuous level, and that our numerical framework satisfies an equivalent balance equation at the discrete level and is genuinely dissipation-free. In the case of ℙ1 Lagrange basis functions and tetrahedral unstructured meshes, a parallel implementation of the method has been developed, based on message passing and mesh partitioning. Three-dimensional numerical results confirm the theoretical properties of the method. They include test-cases where Kelvin–Helmholtz instabilities appear.

APA, Harvard, Vancouver, ISO, and other styles

19

Kulabukhov, S. Yu. "Mathematical modeling in informatiсs lessons using numerical solution of differential equations." Informatics in school, no. 2 (April 27, 2021): 14–21. http://dx.doi.org/10.32517/2221-1993-2021-20-2-14-21.

Full text

Abstract:

The article considers the author's fragment of an in-depth course of informatics in a physics and mathematics school. It is based on the use of differential equations to model complex physical processes. Three models are considered: a mathematical spring pendulum, the orbital motion of the satellite around the planet and the movement of the body in the atmosphere taking into account air resistance. Since in the in-depth course of mathematics of the physics and mathematics school, numerical methods for solving differential equations are not studied, the article proposes to use the simplest method — the Euler method. Using this method, the resulting differential equations in each case are numerically solved. Since differential equations arising in each model have a second order, in order to apply the Euler method, it is necessary to reduce the order of differential equations by introducing new variables. In this case, instead of one differential equation of the second order, a system of two equations of the first order arises. For each built model, its implementation in the PascalABC.NET programming language is given.

APA, Harvard, Vancouver, ISO, and other styles

20

Gavrilyuk, Sergey, and Keh-Ming Shyue. "Hyperbolic approximation of the BBM equation." Nonlinearity 35, no. 3 (February 18, 2022): 1447–67. http://dx.doi.org/10.1088/1361-6544/ac4c49.

Full text

Abstract:

Abstract It is well known that the Benjamin–Bona–Mahony (BBM) equation can be seen as the Euler–Lagrange equation for a Lagrangian expressed in terms of the solution potential. We approximate the Lagrangian by a two-parameter family of Lagrangians depending on three potentials. The corresponding Euler–Lagrange equations can be then written as a hyperbolic system of conservations laws. The hyperbolic BBM system has two genuinely nonlinear eigenfields and one linear degenerate eigenfield. Moreover, it can be written in terms of Riemann invariants. Such an approach conserves the variational structure of the BBM equation and does not introduce the dissipation into the governing equations as it usually happens for the classical relaxation methods. The state-of-the-art numerical methods for hyperbolic conservation laws such as the Godunov-type methods are used for solving the ‘hyperbolized’ dispersive equations. We find good agreement between the corresponding solutions for the BBM equation and for its hyperbolic counterpart.

APA, Harvard, Vancouver, ISO, and other styles

21

Gao, Feng, and Xiao-Jun Yang. "Local fractional Euler’s method for the steady heat-conduction problem." Thermal Science 20, suppl. 3 (2016): 735–38. http://dx.doi.org/10.2298/tsci16s3735g.

Full text

Abstract:

In this paper, the local fractional Euler?s method is proposed to consider the steady heat-conduction problem for the first time. The numerical solution for the local fractional heat-relaxation equation is presented. The comparison between numerical and exact solutions is discussed.

APA, Harvard, Vancouver, ISO, and other styles

22

Abdallah, S., and C. F. Smith. "Three-Dimensional Solutions for Inviscid Incompressible Flow in Turbomachines." Journal of Turbomachinery 112, no. 3 (July 1, 1990): 391–98. http://dx.doi.org/10.1115/1.2927672.

Full text

Abstract:

A primitive variable formulation is used for the solution of the incompressible Euler equation. In particular, the pressure Poisson equation approach using a nonstaggered grid is considered. In this approach, the velocity field is calculated from the unsteady momentum equation by marching in time. The continuity equation is replaced by a Poisson-type equation for the pressure with Neumann boundary conditions. A consistent finite-difference method, which insures the satisfaction of a compatibility condition necessary for convergence, is used in the solution of the pressure equation on a nonstaggered grid. Numerical solutions of the momentum equations are obtained using the second-order upwind differencing scheme, while the pressure Poisson equation is solved using the line successive overrelaxation method. Three turbo-machinery rotors are tested to validate the numerical procedure. The three rotor blades have been designed to have similar loading distributions but different amounts of dihedral. Numerical solutions are obtained and compared with experimental data in terms of the velocity components and exit swirl angles. The computed results are in good agreement with the experimental data.

APA, Harvard, Vancouver, ISO, and other styles

23

Abhyankar, N. S., E. K. Hall, and S. V. Hanagud. "Chaotic Vibrations of Beams: Numerical Solution of Partial Differential Equations." Journal of Applied Mechanics 60, no. 1 (March 1, 1993): 167–74. http://dx.doi.org/10.1115/1.2900741.

Full text

Abstract:

The objective of this paper is to examine the utility of direct, numerical solution procedures, such as finite difference or finite element methods, for partial differential equations in chaotic dynamics. In the past, procedures for solving such equations to detect chaos have utilized Galerkin approximations which reduce the partial differential equations to a set of truncated, nonlinear ordinary differential equations. This paper will demonstrate that a finite difference solution is equivalent to a Galerkin solution, and that the finite difference method is more powerful in that it may be applied to problems for which the Galerkin approximations would be difficult, if not impossible to use. In particular, a nonlinear partial differential equation which models a slender, Euler-Bernoulli beam in compression issolvedto investigate chaotic motions under periodic transverse forcing. The equation, cast as a system of firstorder partial differential equations is directly solved by an explicit finite difference scheme. The numerical solutions are shown to be the same as the solutions of an ordinary differential equation approximating the first mode vibration of the buckled beam. Then rigid stops of finite length are incorporated into the model to demonstrate a problem in which the Galerkin procedure is not applicable. The finite difference method, however, can be used to study the stop problem with prescribed restrictions over a selected subdomain of the beam. Results obtained are briefly discussed. The end result is a more general solution technique applicable to problems in chaotic dynamics.

APA, Harvard, Vancouver, ISO, and other styles

24

Torabi, K., H. Afshari, and E. Zafari. "Transverse Vibration of Non-Uniform Euler-Bernoulli Beam, Using Differential Transform Method (DTM)." Applied Mechanics and Materials 110-116 (October 2011): 2400–2405. http://dx.doi.org/10.4028/www.scientific.net/amm.110-116.2400.

Full text

Abstract:

Analysis of transverse vibration of beams is presented in this paper. Unfortunately, complexities which appear in solving differential equation of transverse vibration of non-uniform beams, limit analytical solution to some special cases, so that the numerical method is presented. DTM is a numerical method for solving linear and some non-linear, ordinary and partial differential equations. In this paper, this technique has been applied for solving differential equation of transverse vibration of conical Euler-Bernoulli beam. Natural circular frequencies and mode shapes have been calculated. Comparing results with the cases which exact solution have been presented, shows that DTM is a strong method especially for solving quasi-linear differential equations.

APA, Harvard, Vancouver, ISO, and other styles

25

Abualhomos, Mayada. "Numerical Solution of Complex Fuzzy Differential Equations by Euler and Taylor Methods." International Journal of Membrane Science and Technology 10, no. 3 (July 20, 2023): 340–53. http://dx.doi.org/10.15379/ijmst.v10i3.1539.

Full text

Abstract:

In 1965, Zadeh introduced the concept of fuzzy set, which is a class of objects with a continuum of grades of membership, such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. In 1987 O. Kaleva defined the concept of fuzzy differential equations and present some basic notions of differential equations such as differentiability, integrability, existence and uniqueness theorem for a solution to a fuzzy differential equation. He also, in 1990 studied the Cauchy problem for fuzzy differential equations and showed that it has a solution if and only if there is a subset and its locally compact. Later, M. Ma, M. Friedman, and A. Kandel 1999 introduced numerical solutions of fuzzy differential equations. In this paper we incorporate the above ideas to introduce numerical solution of complex fuzzy differential equations by Euler and Taylor methods by extending the codomain of membership function of fuzzy topological space from [0, 1] to the unit disk in the complex plane. This extension allows us getting more range and flexibility to represent objects with uncertainty and periodicity semantics without losing the full meaning of information. Also, we considered the definitions of complex fuzzy sets, cartesian and polar representation of complex membership, and Cauchy problem for CFDEs. We then found the exact solutions and approximations for Taylor and Euler methods for CFDEs by levels and where , and provide examples of the results we obtained.

APA, Harvard, Vancouver, ISO, and other styles

26

de Hoog, F. R., and R. S. Anderssen. "Approximate solutions for the Couette viscometry equation." Bulletin of the Australian Mathematical Society 72, no. 3 (December 2005): 461–70. http://dx.doi.org/10.1017/s0004972700035280.

Full text

Abstract:

The recovery of flow curves for non-Newtonian fluids from Couette rheometry measurements involves the solution of a quite simple first kind Volterra integral equation with a discontinuous kernel for which the solution, as a summation of an infinite series, has been known since 1953. Various methods, including an Euler-Maclaurin sum formula, have been proposed for the estimation of the value of the summation. They all involve the numerical differentiation of the observational data. In this paper, the properties of Bernoulli polynomials, in conjunctions with the special structure of the integral equation, are exploited to derive a parametric family of representations for its solution. They yield formulas similar to, but more general than, the previously published Euler-Maclaurin sum formula representations. The parameterisation is then utilised to derive two new classes of approximations. The first yields a family of finite difference approximations, which avoids the direct numerical differentiation of the observational data, while the second generates a framework for the construction of improved power law approximations.

APA, Harvard, Vancouver, ISO, and other styles

27

Ghannadiasl, Amin. "Quintic B-spline collocation method for numerical solution of free vibration of tapered Euler-Bernoulli beam on variable Winkler foundation." Journal of Mechanical Engineering and Sciences 15, no. 2 (June 17, 2021): 8193–204. http://dx.doi.org/10.15282/jmes.15.2.2021.18.0643.

Full text

Abstract:

The collocation method is the method for the numerical solution of integral equations and partial and ordinary diﬀerential equations. The main idea of this method is to choose a number of points in the domain and a ﬁnite-dimensional space of candidate solutions. So, that solution satisﬁes the governing equation at the collocation points. The current paper involves developing, and a comprehensive, step-by step procedure for applying the collocation method to the numerical solution of free vibration of tapered Euler-Bernoulli beam. In this stusy, it is assumed the beam rested on variable Winkler foundation. The simplicity of this approximation method makes it an ideal candidate for computer implementation. Finally, the numerical examples are introduced to show eﬃciency and applicability of quintic B-spline collocation method. Numerical results are demonstrated that quintic B-spline collocation method is very competitive for numerical solution of frequency analysis of tapered beam on variable elastic foundation.

APA, Harvard, Vancouver, ISO, and other styles

28

Xu, Xiaoyong, and Fengying Zhou. "Orthonormal Euler wavelets method for time-fractional Cattaneo equation with Caputo-Fabrizio derivative." AIMS Mathematics 8, no. 2 (2022): 2736–62. http://dx.doi.org/10.3934/math.2023144.

Full text

Abstract:

<abstract><p>In this paper, a new orthonormal wavelets based on the orthonormal Euler polynomials (OEPs) is constructed to approximate the numerical solution of time-fractional Cattaneo equation with Caputo-Fabrizio derivative. By applying the Gram-Schmidt orthonormalization process on sets of Euler polynomials of various degrees, an explicit representation of OEPs is obtained. The convergence analysis and error estimate of the orthonormal Euler wavelets expansion are studied. The exact formula of Caputo-Fabrizio fractional integral of orthonormal Euler wavelets are derived using Laplace transform. The applicability and validity of the proposed method are verified by some numerical examples.</p></abstract>

APA, Harvard, Vancouver, ISO, and other styles

29

Shiea, Mohsen, Antonio Buffo, Marco Vanni, and Daniele Marchisio. "Numerical Methods for the Solution of Population Balance Equations Coupled with Computational Fluid Dynamics." Annual Review of Chemical and Biomolecular Engineering 11, no. 1 (June 7, 2020): 339–66. http://dx.doi.org/10.1146/annurev-chembioeng-092319-075814.

Full text

Abstract:

This review article discusses the solution of population balance equations, for the simulation of disperse multiphase systems, tightly coupled with computational fluid dynamics. Although several methods are discussed, the focus is on quadrature-based moment methods (QBMMs) with particular attention to the quadrature method of moments, the conditional quadrature method of moments, and the direct quadrature method of moments. The relationship between the population balance equation, in its generalized form, and the Euler-Euler multiphase flow models, notably the two-fluid model, is thoroughly discussed. Then the closure problem and the use of Gaussian quadratures to overcome it are analyzed. The review concludes with the presentation of numerical issues and guidelines for users of these modeling approaches.

APA, Harvard, Vancouver, ISO, and other styles

30

Blaszczyk, Tomasz. "Analytical and numerical solution of the fractional Euler–Bernoulli beam equation." Journal of Mechanics of Materials and Structures 12, no. 1 (January 1, 2017): 23–34. http://dx.doi.org/10.2140/jomms.2017.12.23.

Full text

APA, Harvard, Vancouver, ISO, and other styles

31

Khalili Golmankhaneh, Alireza, and Carlo Cattani. "Fractal Logistic Equation." Fractal and Fractional 3, no. 3 (July 11, 2019): 41. http://dx.doi.org/10.3390/fractalfract3030041.

Full text

Abstract:

In this paper, we give difference equations on fractal sets and their corresponding fractal differential equations. An analogue of the classical Euler method in fractal calculus is defined. This fractal Euler method presets a numerical method for solving fractal differential equations and finding approximate analytical solutions. Fractal differential equations are solved by using the fractal Euler method. Furthermore, fractal logistic equations and functions are given, which are useful in modeling growth of elements in sciences including biology and economics.

APA, Harvard, Vancouver, ISO, and other styles

32

Basirat, Behrooz, and Hamid Reza Elahi. "An Improved Collocation Approach of Euler Polynomials Connected with Bernoulli Ones for Solving Predator-Prey Models with Time Lag." International Journal of Differential Equations 2020 (April 1, 2020): 1–8. http://dx.doi.org/10.1155/2020/9176784.

Full text

Abstract:

This paper deals with an approach to obtaining the numerical solution of the Lotka–Volterra predator-prey models with discrete delay using Euler polynomials connected with Bernoulli ones. By using the Euler polynomials connected with Bernoulli ones and collocation points, this method transforms the predator-prey model into a matrix equation. The main characteristic of this approach is that it reduces the predator-prey model to a system of algebraic equations, which greatly simplifies the problem. For these models, the explicit formula determining the stability and the direction is given. Numerical examples illustrate the reliability and efficiency of the proposed scheme.

APA, Harvard, Vancouver, ISO, and other styles

33

BUCKWAR, E., M. G. RIEDLER, and P. E. KLOEDEN. "THE NUMERICAL STABILITY OF STOCHASTIC ORDINARY DIFFERENTIAL EQUATIONS WITH ADDITIVE NOISE." Stochastics and Dynamics 11, no. 02n03 (September 2011): 265–81. http://dx.doi.org/10.1142/s0219493711003279.

Full text

Abstract:

An asymptotic stability analysis of numerical methods used for simulating stochastic differential equations with additive noise is presented. The initial part of the paper is intended to provide a clear definition and discussion of stability concepts for additive noise equation derived from the principles of stability analysis based on the theory of random dynamical systems. The numerical stability analysis presented in the second part of the paper is based on the semi-linear test equation dX(t) = (AX(t) + f(X(t))) dt + σ dW(t), the drift of which satisfies a contractive one-sided Lipschitz condition, such that the test equation allows for a pathwise stable stationary solution. The θ-Maruyama method as well as linear implicit and two exponential Euler schemes are analysed for this class of test equations in terms of the existence of a pathwise stable stationary solution. The latter methods are specifically developed for semi-linear problems as they arise from spatial approximations of stochastic partial differential equations.

APA, Harvard, Vancouver, ISO, and other styles

34

Ahmad, Imtiaz, Sayed Abdel-Khalek, Ahmed Alghamdi, and Mustafa Inc. "Numerical simulation of the generalized Burger’s-Huxley equation via two meshless methods." Thermal Science 26, Spec. issue 1 (2022): 463–68. http://dx.doi.org/10.2298/tsci22s1463a.

Full text

Abstract:

Numerical solution of the generalized Burger?s-Huxley equation is established utilizing two effective meshless methods namely: local differential quadrature method and global method of line. Both the proposed meshless methods used radial basis functions to discretize space derivatives which convert the given model equation system of ODE and then we have utilized the Euler method to get the required numerical solution. Numerical experiments are carried out to check the efficiency and accuracy of the suggested meshless methods.

APA, Harvard, Vancouver, ISO, and other styles

35

WANG, CHUN-WEI, and ZVI RUSAK. "Numerical studies of transonic BZT gas flows around thin airfoils." Journal of Fluid Mechanics 396 (October 10, 1999): 109–41. http://dx.doi.org/10.1017/s0022112099005893.

Full text

Abstract:

Numerical studies of two-dimensional, transonic flows of dense gases of retrograde type, known as BZT gases, around thin airfoils are presented. The computations are guided by a recent asymptotic theory of Rusak & Wang (1997). It provides a uniformly valid solution of the flow around the entire airfoil surface which is composed of outer and inner solutions. A new transonic small-disturbance (TSD) equation solver is developed to compute the nonlinear BZT gas flow in the outer region around most of the airfoil. The flow in the inner region near the nose of the airfoil is computed by solving the problem of a sonic flow around a parabola. Numerical results of the composite solutions calculated from the asymptotic formula are compared with the solutions of the Euler equations. The comparison demonstrates that, in the leading order, the TSD solutions of BZT gas flows represent the essence of the flow character around the airfoil as computed from the Euler equations. Furthermore, guided by the asymptotic formula, the computational results demonstrate the similarity rules for transonic flows of BZT gases. There are differences between the self-similar cases that may be related to the error associated with the accuracy of the asymptotic solution. A discussion on the flow patterns around an airfoil at transonic speeds and at various upstream thermodynamic conditions is also presented. The paper provides important guidelines for future studies on this subject.

APA, Harvard, Vancouver, ISO, and other styles

36

Lanthaler, Samuel, and Siddhartha Mishra. "Computation of measure-valued solutions for the incompressible Euler equations." Mathematical Models and Methods in Applied Sciences 25, no. 11 (July 10, 2015): 2043–88. http://dx.doi.org/10.1142/s0218202515500529.

Full text

Abstract:

We combine the spectral (viscosity) method and ensemble averaging to propose an algorithm that computes admissible measure-valued solutions of the incompressible Euler equations. The resulting approximate young measures are proved to converge (with increasing numerical resolution) to a measure-valued solution. We present numerical experiments demonstrating the robustness and efficiency of the proposed algorithm, as well as the appropriateness of measure-valued solutions as a solution framework for the Euler equations. Furthermore, we report an extensive computational study of the two-dimensional vortex sheet, which indicates that the computed measure-valued solution is non-atomic and implies possible non-uniqueness of weak solutions constructed by Delort.

APA, Harvard, Vancouver, ISO, and other styles

37

Zeng, Shijie, and Yaqing Liu. "The Whitham Modulation Solution of the Complex Modified KdV Equation." Mathematics 11, no. 13 (June 22, 2023): 2810. http://dx.doi.org/10.3390/math11132810.

Full text

Abstract:

This paper primarily concerns the Whitham modulation equation of the complex modified Korteweg–de Vries (cmKdV) equation with a step-like initial value. By utilizing the Lax pair, we derive the N-genus Whitham equations via the averaging method. The Whitham equation can be integrated using the hodograph transformation. We investigate Krichever’s algebro-geometric scheme to propose the averaging method for the cmKdV integrable hierarchy and obtain the Whitham velocities of the integrable hierarchy and the hodograph transformation. The connection between the equations of the Euler–Poisson–Darboux type linear overdetermined system, which determines the solutions of the hodograph transformation, is constructed through Riemann integration, which demonstrates that the Whitham equation can be solved. Finally, a step-like initial value problem is solved and an exotic wave pattern is discovered. The results of direct numerical simulation agree well with the Whitham theory solution, which shows the validity of the theory.

APA, Harvard, Vancouver, ISO, and other styles

38

Zhao, Wen-bin, Zhi-xin Hu, Xue-xia Zhang, and Hai-ling Xie. "The Static Stress Intensity Factor around the Antiplane Crack in an Infinite FGM Strip." Mathematical Problems in Engineering 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/628934.

Full text

Abstract:

The problem of the static stress intensity factor around antiplane crack in an infinite strip functionally graded material was studied by using the method of integral transform-dual integral equations in this paper. The shear modulus in the two principal directions of the functionally graded material was assumed to vary proportionately as gradient model of double parameters index function. The partial differential equation was first reduced to Euler equation with Fourier cosine transform. By solving dual integral equations that were derived by applying the solution of Euler equation with the method of Copson, stress intensity factor around the crack tip was derived. And the variation curves of the dimensionless stress intensity factor with the strip height, crack length, gradient parameter, and inhomogeneous coefficient are obtained by using the numerical calculation.

APA, Harvard, Vancouver, ISO, and other styles

39

Tian, Zhikun, Yanping Chen, and Jianyun Wang. "Two-Grid Method for a Fully Discrete Mixed Finite Element Solution of the Time-Dependent Schrödinger Equation." Mathematics 11, no. 14 (July 15, 2023): 3127. http://dx.doi.org/10.3390/math11143127.

Full text

Abstract:

We study the backward Euler fully discrete mixed finite element method for the time-dependent Schrödinger equation; the error result of the mixed finite element solution is obtained in the L2-norm with order O(τ+hk+1). Then, a two-grid method is presented with a backward Euler fully discrete scheme. Using this method, we solve the original problem on a much coarser grid and solve elliptic equations on a fine grid. In addition, the error of the two-grid solution is also obtained in the L2-norm with order O(τ+hk+1+Hk+2). The numerical experiment is provided to demonstrate the efficiency of the algorithm.

APA, Harvard, Vancouver, ISO, and other styles

40

He, L., and J. D. Denton. "Inviscid-Viscous Coupled Solution for Unsteady Flows Through Vibrating Blades: Part 1—Description of the Method." Journal of Turbomachinery 115, no. 1 (January 1, 1993): 94–100. http://dx.doi.org/10.1115/1.2929222.

Full text

Abstract:

An efficient coupled approach between inviscid Euler and integral boundary layer solutions has been developed for quasi-3-D unsteady flows induced by vibrating blades. For unsteady laminar and turbulent boundary layers, steady correlations are adopted in a quasi-steady way to close the integral boundary layer model. This quasi-steady adoption of the correlations is assessed by numerical test results using a direct solution of the unsteady momentum integral equation. To conduct the coupling between the inviscid and viscous solutions for strongly interactive flows, the unsteady Euler and integral boundary layer equations are simultaneously time-marched using a multistep Runge–Kutta scheme, and the boundary layer displacement effect is accounted for by a first order transpiration model. This time-resolved coupling method converges at conditions with considerable boundary layer separation.

APA, Harvard, Vancouver, ISO, and other styles

41

Koroglu, Canan, and Ayhan Aydin. "An Unconventional Finite Difference Scheme for Modified Korteweg-de Vries Equation." Advances in Mathematical Physics 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/4796070.

Full text

Abstract:

A numerical solution of the modified Korteweg-de Vries (MKdV) equation is presented by using a nonstandard finite difference (NSFD) scheme with theta method which includes the implicit Euler and a Crank-Nicolson type discretization. Local truncation error of the NSFD scheme and linear stability analysis are discussed. To test the accuracy and efficiency of the method, some numerical examples are given. The numerical results of NSFD scheme are compared with the exact solution and a standard finite difference scheme. The numerical results illustrate that the NSFD scheme is a robust numerical tool for the numerical integration of the MKdV equation.

APA, Harvard, Vancouver, ISO, and other styles

42

Eddahbi, Mhamed, Lassaad Mchiri, and Mohamed Rhaima. "Numerical solution of quadratic SDE with measurable drift." Filomat 36, no. 15 (2022): 5263–78. http://dx.doi.org/10.2298/fil2215263e.

Full text

Abstract:

In this paper we are interested in solving numerically quadratic SDEs with non-necessary continuous drift of the from Xt = x + ? t 0 b(s,Xs)ds + ? t 0 f (Xs)?2(Xs)ds + ? t 0 ?(Xs)dWs, where, x is the initial data b and ? are given coefficients that are assumed to be Lipschitz and bounded and f is a measurable bounded and integrable function on the whole space R. Numerical simulations for this class of SDE of quadratic growth and measurable drift, induced by the singular term f (x)?2(x), is implemented and illustrated by some examples. The main idea is to use a phase space transformation to transform our initial SDEs to a standard SDE without the discontinuous and quadratic term. The Euler-Maruyama scheme will be used to discretize the new equation, thus numerical approximation of the original equation is given by taking the inverse of the space transformation. The rate of convergence are shown to be of order 1/2.

APA, Harvard, Vancouver, ISO, and other styles

43

Torabi, Mina, Manuel Pastor, and Miguel Martín Stickle. "Three-step Predictor-Corrector Finite Element Schemes for Consolidation Equation." Mathematical Problems in Engineering 2020 (April 28, 2020): 1–14. http://dx.doi.org/10.1155/2020/2873869.

Full text

Abstract:

An accurate, stable, and efficient three-step predictor-corrector time integration method is considered, for the first time, to obtain numerical solution for the one-dimensional consolidation equation within a finite and spectral element framework. Theoretical order of accuracy and stability conditions are provided. The three-step predictor-corrector time integration method is third-order accurate and shows a larger stability region than the forward Euler method when applied to the one-dimensional consolidation equation. Furthermore, numerical results are in agreement with analytical solutions previously derived by the authors.

APA, Harvard, Vancouver, ISO, and other styles

44

Kolarova, Edita, and Lubomir Brancik. "Confidence intervals for RLCG cell influenced by coloured noise." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 36, no. 4 (July 3, 2017): 838–49. http://dx.doi.org/10.1108/compel-07-2016-0321.

Full text

Abstract:

Purpose The purpose of this paper is to determine confidence intervals for the stochastic solutions in RLCG cells with a potential source influenced by coloured noise. Design/methodology/approach The deterministic model of the basic RLCG cell leads to an ordinary differential equation. In this paper, a stochastic model is formulated and the corresponding stochastic differential equation is analysed using the Itô stochastic calculus. Findings Equations for the first and the second moment of the stochastic solution of the coloured noise-affected RLCG cell are obtained, and the corresponding confidence intervals are determined. The moment equations lead to ordinary differential equations, which are solved numerically by an implicit Euler scheme, which turns out to be very effective. For comparison, the confidence intervals are computed statistically by an implementation of the Euler scheme using stochastic differential equations. Practical implications/implications The theoretical results are illustrated by examples. Numerical simulations in the examples are carried out using Matlab. A possible generalization for transmission line models is indicated. Originality/value The Itô-type stochastic differential equation describing the coloured noise RLCG cell is formulated, and equations for the respective moments are derived. Owing to this original approach, the confidence intervals can be found more effectively by solving a system of ordinary differential equations rather than by using statistical methods.

APA, Harvard, Vancouver, ISO, and other styles

45

Tekin, Ibrahim, and He Yang. "INVERSE PROBLEM FOR THE TIME-FRACTIONAL EULER-BERNOULLI BEAM EQUATION." Mathematical Modelling and Analysis 26, no. 3 (September 10, 2021): 503–18. http://dx.doi.org/10.3846/mma.2021.13289.

Full text

Abstract:

In this paper, the classical Euler-Bernoulli beam equation is considered by utilizing fractional calculus. Such an equation is called the time-fractional EulerBernoulli beam equation. The problem of determining the time-dependent coefficient for the fractional Euler-Bernoulli beam equation with homogeneous boundary conditions and an additional measurement is considered, and the existence and uniqueness theorem of the solution is proved by means of the contraction principle on a sufficiently small time interval. Numerical experiments are also provided to verify the theoretical findings.

APA, Harvard, Vancouver, ISO, and other styles

46

Kazakov, Alexander, and Lev Spevak. "Constructing Exact and Approximate Diffusion Wave Solutions for a Quasilinear Parabolic Equation with Power Nonlinearities." Mathematics 10, no. 9 (May 5, 2022): 1559. http://dx.doi.org/10.3390/math10091559.

Full text

Abstract:

The paper studies a degenerate nonlinear parabolic equation containing a convective term and a source (reaction) term. It considers the construction of approximate solutions to this equation with a specified law of diffusion wave motion, the existence of these solutions being proved in our previous studies. A stepwise algorithm of the numerical solution with a time-difference scheme is proposed, the second-order difference scheme being used in such problems for the first time. At each step the problem is solved iteratively on the basis of a radial basis function (RBF) collocation method. In order to verify the numerical solution algorithm, two classes of exact generalized traveling wave solutions are proposed, whose construction is reduced to solving a Cauchy problem for second order ordinary differential equations (ODEs) with a singularity at the higher derivative. The theorem of the existence and uniqueness of the analytical solution in the form of a power series is proved for it, and the estimates of the radius of convergence are obtained. The Euler method is used to prove a similar statement concerning the existence of a continuous solution in the non-analytical case. The RBF collocation method is also applied for the approximate solution of the Cauchy problem. The solutions to the Cauchy problem are numerically analyzed, and this has enabled us to reveal and describe some of their properties, including those not previously observed, and to assess the accuracy of the method.

APA, Harvard, Vancouver, ISO, and other styles

47

Potra, Florian A., and Werner C. Rheinbold. "On the Numerical Solution of Euler-Lagrange Equations∗." Mechanics of Structures and Machines 19, no. 1 (January 1991): 1–18. http://dx.doi.org/10.1080/08905459108905135.

Full text

APA, Harvard, Vancouver, ISO, and other styles

48

Lin, Yanping, Shuzhan Xu, and Hong-Ming Yin. "Finite difference approximations for a class of non-local parabolic equations." International Journal of Mathematics and Mathematical Sciences 20, no. 1 (1997): 147–63. http://dx.doi.org/10.1155/s0161171297000215.

Full text

Abstract:

In this paper we study finite difference procedures for a class of parabolic equations with non-local boundary condition. The semi-implicit and fully implicit backward Euler schemes are studied. It is proved that both schemes preserve the maximum principle and monotonicity of the solution of the original equation, and fully-implicit scheme also possesses strict monotonicity. It is also proved that finite difference solutions approach to zero ast→∞exponentially. The numerical results of some examples are presented, which support our theoretical justifications.

APA, Harvard, Vancouver, ISO, and other styles

49

Biswas, Swapan, and Uttam Ghosh. "Approximate Solution of Homogeneous and Nonhomogeneous 5αth-Order Space-Time Fractional KdV Equations." International Journal of Computational Methods 18, no. 01 (June 30, 2020): 2050018. http://dx.doi.org/10.1142/s0219876220500188.

Full text

Abstract:

In this paper, the semi-inverse method is applied to derive the Lagrangian of the [Formula: see text]th Korteweg de Vries equation (KdV). Then the time and space differential operators of the Lagrangian are replaced by corresponding fractional derivatives. The variation of the functional of this Lagrangian is devoted to lead the fractional Euler Lagrangian via Agrawal’s method, which gives the space-time fractional KdV equation. Jumarie derivative is used to obtain the space-time fractional KdV equations. The homotopy analysis method (HAM) is applied to solve the derived space-time fractional KdV equation. Then numerical solutions are compared with the known analytical solutions by tables and figures.

APA, Harvard, Vancouver, ISO, and other styles

50

Adair, Desmond, and Martin Jaeger. "A power series solution for rotating nonuniform Euler–Bernoulli cantilever beams." Journal of Vibration and Control 24, no. 17 (June 14, 2017): 3855–64. http://dx.doi.org/10.1177/1077546317714183.

Full text

Abstract:

A systematic procedure is developed for studying the dynamic response of a rotating nonuniform Euler–Bernoulli beam with an elastically restrained root. To find the solution, a novel approach is used in that the fourth-order differential equation describing the vibration problem is first written as a first-order matrix differential equation, which is then solved using the power series method. The method can be used to obtain an approximate solution of vibration problems for nonuniform Euler–Bernoulli beams. Specifically, numerical examples are presented here to demonstrate the usefulness of the method in frequency analysis of nonuniform Euler–Bernoulli clamped-free cantilever beams. Results for mode shapes and frequency parameters were found to be in satisfactory agreement with previously published results. The effects of tapering, both equal and unequal, were investigated for both a cantilever wedge and cantilever cone.

APA, Harvard, Vancouver, ISO, and other styles

Journal articles on the topic 'Euler Equation - Numerical Solution'

Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles